Implications of some less-known scattering approximations in a deuteron-model bound state

We compare results obtained in potential-model testing for the³S bound state of the deuteron by some scattering approximations which are not familiar in literature. The test concerns: the plane-wave approximation used in the continuous-energy spectrum within the framework of the Gelfand-Levitan formalism, the Brysk approximation modified byMichalík (Czech. J. Phys.B 17 (1967), 757;B 18 (1968), 517) and the Jost method modified bySasakawa (Prog. Theoret. Phys. Suppl. No. 27 (1963), 1). It is pointed out which of these approximations can be preferred in similar model calculations, where the wave-function approach is demanded. In particular cases we also give contributions for the first-order energy shift due to a weak Yukawa perturbation on the deuteron, if its bound state is described by the mentioned approximative wave functions.     

Time-dependent density functional calculation of e-H scattering

Phase shifts for single-channel elastic electron-atom scattering are derived from time-dependent density functional theory. The H- ion is placed in a spherical box, its discrete spectrum found, and phase shifts deduced. Exact exchange yields an excellent approximation to the ground-state Kohn-Sham potential, while the adiabatic local density approximation yields good singlet and triplet phase shifts.     

Analytical formulae for the optimization of the process of low-power phase modulation in a quadratic nonlinear medium

We show that using the approximation of fixed intensity analytical formulae, describing the process of induced phase modulation for the beams involved in second-order nonlinear optical processes can be derived. Expressions that allow the optimization of the phase shifts experienced by the fundamental and generated waves are presented for nonlinear quadratic processes, second-harmonic generation and sum-frequency mixing. In the case of seeding at the generated wavelength, the phase shift of the fundamental wave is due to two interactions: (i) a cubic one, based on coupled second-order processes (cascade cubic nonlinearity) and (ii) single quadratic interaction with participation of the seeding wave. By comparison with the exact numerical solution, we defined the input parameters of the beams for which this analytical approach is valid. It is shown that phase shifts exceeding /2 can be correctly predicted using the expressions obtained.     

Non-Relativistic Phase Shifts for Scattering on Generalized Radial Yukawa Potentials

Non-relativistic phase shifts for a generalized Yukawa potential V(r) =-V_0( e~(-αr)/r)-V_1( e~(-2αr)/r~2) are studied by the amplitude-phase method and by a frequently used analytic method based on a Pekeris-type approximation of power-law potential terms.Small variations of V_1 seem to have marginal effects on the effective potential and on exact phase shifts.However,as pointed out in this study,a Pekeris-type approximation in scattering applications often implies serious distortions of both effective potentials and phase shifts.The Pekeris-type based analytic approximation in this study seems to give low-quality scattering results for this model potential at low energies.     

Inclusion of Final-State-Interaction Effects at Higher Energies in the Photodisintegration of the α-Particle

Final-state-interaction (FSI) effects are included in the treatment of the photodisintegration of the α-particle via the K-matrix approximation. These are applied to the exact AGS-type integral equation for this process at photon energies 50–100 MeV. Direct comparison with the plane-wave (Born) approximation indicates that these effects are small, which explains the good fit to the data we found previously. The FSI in the integro-differential equation treatment of the problem in configuration space are included by means of effective nucleon-trinucleon potentials constructed via an exact inverse scattering method and by using the phase shifts obtained from the K-matrix approach to the 3 + 1 → 3 + 1 scattering. Comparison between this approach and the integral-equation approach confirms the validity of this model previously employed at low energies using experimental phase shifts. Finally we show that the photonuclear cross sections are sensitive to the details of the boundstate wave functions of the ³He and ⁴He as calculated exactly via the AGS equations and in the integro-differential-equation approach, which for the ⁴He neglects higher-order correlations. Received April 18, 1994; revised September 12, 1994; accepted for publication October 12, 1994     

Continuum states of modified Morse potential

Using a proper approximation scheme to the centrifugal term, we study any l-wave continuum states of the Schrdinger equation for the modified Morse potential. The normalised analytical radial wave functions are presented, and a corresponding calculation formula of phase shifts is derived. It is shown that the energy levels of the continuum states reduce to those of the bound states at the poles of the scattering amplitude. Some numerical results are calculated to show the accuracy of our results.     

Scattering states of modified Pöschlben Teller potential in D-dimension

We present a new approximation scheme for the centrifugal term, and apply this new approach to the Schrödinger equation with the modified Pöschl-Teller potential in the D-dimension for arbitrary angular momentum states. The approximate analytical solutions of the scattering states are derived. The normalized wave functions expressed in terms of the hypergeometric functions of the scattering states on the k/2π scale and the calculation formula of the phase shifts are given. The numerical results show that our results are in good agreement with those obtained by using the amplitude-phase method (APM).     

Approximate analytical solutions of scattering states for Klein-Gordon equation with Hulthén potentials for nonzero angular momentum

In this paper, using the exponential function transformation approach along with an approximation for the centrifugal potential, the radial Klein-Gordon equation with the vector and scalar Hulthén potential is transformed to a hypergeometric differential equation. The approximate analytical solutions of t-waves scattering states are presented. The normalized wave functions expressed in terms of hypergeometric functions of scattering states on the “k/2π scale” and the calculation formula of phase shifts are given. The physical meaning of the approximate analytical solution is discussed.      

The relativistic bound and scattering states of the Manning-Rosen potential with an improved new approximate scheme to the centrifugal term

The approximately analytical bound and scattering state solutions of the arbitrary l-wave Klein-Gordon equation for the mixed Manning-Rosen potentials are carried out by an improved new approximation to the centrifugal term. The normalized analytical radial wave functions of the l-wave Klein-Gordon equation with the mixed Manning-Rosen potentials are presented and the corresponding energy equations for bound states and phase shifts for scattering states are derived. It is shown that the energy levels of the continuum states, reduce to the bound states of those at the poles of the scattering amplitude. Some useful figures are plotted to show the improved accuracy of our results and the special case for wave is studied briefly.      

Multiple scattering with a linearized propagator

Scattering by a many-body system is studied within the framework of the “fixed scatterer” approximation and the eikonal approximation formulated in terms of a linearized propagator. If properly treated, the “fixed scatterer” approximation is able to take into account the center-of-mass motion. We specifically study the linearized propagator proposed by Abarbanel and Itzykson. Although for potential scattering the above approximation is essentially equivalent to the Glauber eikonal approximation, its physical implications are quite different when applied to scattering by a composite system. The multiple-scattering series can generally no longer be simply expressed in terms of the individual on-shell scattering amplitudes, and the additivity of phase shifts is shown to break down for overlapping potentials. The implications for phenomenological calculations are discussed. Finally, the above approximation is explicitly applied to high-energy elastic nucleon-deuteron scattering and the results are compared with several variants of the Glauber multiple-scattering formalism.     

Approximation of phase shifts with cut-off parameter

A new method for the approximate calculation of phase shifts is presented. It divides the potential into an inner and outer region as in Swan's method, but has the advantage of not using any form factor. On the other hand, it is inapplicable for energies in the neighbourhood of zero points of the functionj _l (k). The approximation is compared with previous ones for the potential well (barrier) and s-wave scattering.The author is indebted to Dr. J. Fischer CSc. for valuable comments.     

Phase shift calculation for low-energy electron scattering from mercury

Summary The relativistic formalism is used to study the scattering of electrons from mercury at low energy. A method of numerical integration is applied to calculate phase shifts by including a local exchange approximation and phenomenological polarization potential. The phase shifts are found to exhibit a resonant behaviour which is in conformity with the results of some earlier calculations. In contrast to the previous methods, this approach has a general validity over the entire energy range and is free from divergences. The author of this paper has agreed to not receive proofs for correction.     

Dynamical effects of QCD in q2$ \bar{{q}}^{{2}}_{}$ systems

We study the coupling of a tetraquark system to an exchanged meson-meson channel, using a pure gluonic theory based four-quark potential matrix model which is known to fit well a large number of data points for lattice simulations of different geometries of a four-quark system. We find that if this minimal-area-based potential matrix replaces the earlier used simple Gaussian form for the gluon field overlap factor f in its off-diagonal terms, the resulting T -matrix and phase shifts develop an angle dependence whose partial-wave analysis reveals D wave and higher angular-momentum components in it. In addition to the obvious implications of this result for the meson-meson scattering, this new feature indicates the possibility of orbital excitations influencing properties of meson-meson molecules through a polarization potential. We have used a formalism of the resonating group method, treated kinetic energy and overlap matrices on model of the potential matrix, but decoupled the resulting complicated integral equations through the Born approximation. In this exploratory study we have used a quadratic confinement and not included the spin dependence; we also used the approximation of equal constituent quark masses.     

Ionization of ytterbium atoms from an excited state

The cross sections of the photoionization and the electron impact-induced ionization of Yb atoms from the excited 6s6p(³ P ₁) state are numerically calculated. Matrix elements are computed in multielectron relativistic and nonrelativistic approximations with allowance for the superposition of configurations and a relaxation effect. The radial part of the electron wavefunction in a continuous spectrum is calculated using the solutions to one-configuration Hartree-Fock and Dirac-Fock equations. The cross sections calculated by a relativistic method are compared to those for a nonrelativistic approximation. The ratios of the radiation reduced matrix elements and the phase shifts of the wavefunctions of a continuous spectrum calculated for the 6p ɛs and 6p → ɛd transitions are compared to the values obtained by approximating the experimental dependences of the angular distribution of photoelectrons for the photoionization by ultraviolet radiation from an oriented excited state.     

Exact solution of the Dirac equation with a central potential

The exact solution of the Dirac equation with a central potential, in the semi-relativistic approximation, is derived and formulae for phase shifts and eigenvalue equations are given.     

Modelling of a displacive transformation in two-dimensional system within a single-site approximation of continuous displacement cluster variation method

It is attempted to model a displacive phase transformation by Continuous Displacement Cluster Variation Method. The main focus of the present study is placed on distorted to non-distorted phase transformation in the two-dimensional square lattice. The entropy is formulated within the single-site approximation (point approximation), while the pair-wise atomic interaction energies are combined to stabilize a distorted phase at low temperatures. The distorted to non-distorted phase transformation in the present study is of the second order, and the calculated distribution of atoms suggests that the transformation is of the displacive type in the classification.     

Lateral shift of X-ray beams and determination of phase in reflectometry of multilayer periodic structures

Lateral shift of X-ray beams during Bragg’s reflection from a multilayer periodic structure (MS) is studied analytically and numerically. The field distribution in the MS, as well as the displacement of reflected and transmitted beams are determined. Analytic expressions for the shifts are derived in the approximation of spectrally narrow beams. Since the shift is controlled by the phase of the reflectance (transmittance), it is possible in principle to extract information concerning the phase of reflection (transmission) by measuring the spatial shift.     

Variable amplitude equations for one-dimensional scattering

Nonlinear first-order equations, similar to Calogero's equations, are derived for the forward and backward one-dimensional scattering amplitudes. In particular, the even potential case yields two uncoupled equations for the even and odd parity phase shifts. The present approach provides a fast and accurate means for the numerical solution of one-dimensional scattering problems. It also has many analytic merits, some of which are discussed. The connection between one-dimensional and three-dimensional high-energy scattering is reviewed. It is demonstrated that in the one-dimensional case, a slightly modified WKB wavefunction provides an excellent approximation to the exact wavefunction in the shortwave limit. In this limit, additivity of phase shifts for nonoverlapping static potentials is satisfied.     

Interaction of a two level atom with many photons

The interaction of a two level atom with many photons in a continuum of modes is investigated on the base of a Weisskopf-Wigner theory of infinite order. Special emphasis is given to the problem whether a given uncertainty in the number of incident photons has observable effects upon the atom. This refers in particular to the special uncertainty related to the Poisson distribution of photons in a fully coherent state. It is shown that, if at all, the photon number uncertainty can only shift the atomic levels. These shifts are of the order of magnitude of the 2S _1/2–2P _1/2 level separation by Lambschifts. In an approximation in which these level shifts are omitted the photon number uncertainty has no effects upon the atom: The atom interacts simultaneously withall n-photon components of the incident beam, but inindependent interaction processes taking place in orthogonal Hilbert-spaces. Arguments are given to justify the mentioned approximation, a variant of the usual rotating wave approximation. This approximation reduces the Weisskopf-Wigner theory of infinite order to an infinite set of Weisskopf-Wigner theories of finite, lowest order. The latter govern the independent processes mentioned.